Question

Show that $cos^{2}(\frac{\pi}{10}) + cos^{2}(\frac{2\pi}{5}) + cos^{2}(\frac{3\pi}{5}) + cos^{2}(\frac{9\pi}{10})$ = 2.

Answer 1

Hi there!
Here's the answer:

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¶ POINTS TO REMEMBER :

$cos^{2} x = sin (\frac{\pi}{2} - x)$

$sin^{2} x + cos^{2} x = 1$

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¶¶¶ SOLUTION :

$LHS\: =$

= $cos^{2} (\frac{\pi}{10}) + cos^{2} (\frac{2\pi}{5}) + cos^{2} (\frac{3\pi}{5}) + cos^{2} (\frac9{\pi}{10})$

Rearrange terms as shown below

= $cos^{2} (\frac{\pi}{10}) + cos^{2} (\frac{3\pi}{5}) + cos^{2} (\frac{2\pi}{5}) + cos^{2} (\frac9{\pi}{10})$

= $cos^{2} (\frac{\pi}{10}) +[sin (\frac{\pi}{2} - \frac{3\pi}{5})]^{2} + cos^{2} (\frac{2\pi}{5}) + [sin (\frac{\pi}{2} - \frac{9\pi}{10})]^{2}$

= $cos^{2} (\frac{\pi}{10}) +[sin (\frac{-\pi}{10})]^{2} + cos^{2} (\frac{2\pi}{5}) + [sin (\frac{\pi}{2} - \frac{9\pi}{10})]^{2}$

= $cos^{2} (\frac{\pi}{10}) +[sin (\frac{\pi}{10})]^{2} + cos^{2} (\frac{2\pi}{5}) + [sin (\frac{5\pi - 9\pi}{10})]^{2}$

= $cos^{2} (\frac{\pi}{10}) +sin^{2} (\frac{\pi}{10}) + cos^{2} (\frac{2\pi}{5}) + sin (\frac{4\pi}{10})^{2}$

= $[cos^{2} (\frac{\pi}{10}) +sin^{2} (\frac{\pi}{10})] + [cos^{2} (\frac{2\pi}{5}) + sin^{2} (\frac{2\pi}{5})]$

= 1 + 1

= 2

$=\: RHS$

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